 Thank you, Keith, and thank you to the organizers, organizers for the invitation. It's always great to be here in Trieste, and better if there is a great conference like this. And before I forget, this is a joint work with Carlos Vázquez with this, Vázquez, from the Catholic University of Valparaiso, Chile. Okay, I will work with the three torus, a partially hyperbolic defiomorphism. That means, almost everybody knows now, but that the tangent bundle splits into three invariant bundles. In this case, the three bundles are one-dimensional, and this is a contracted bundle, this is a spanding, and the center bundle has an intermediate behavior. And then I will talk about attractors in this setting. The question we have with Carlos is if it is possible to have two attractors with intermingled basins in a robust way. Well, I will show that in this setting it's impossible to have this, but there are examples. I will show you these kind of examples, but I will start with a very simple example that is the following. You take your favorite linear hyperbolic automorphism of the two torus, two and one-one, something like this, and take this defium that is A times some north pole, south pole of the circle. This is from T2 times S1 to S1, and then you have the following situation. We have the one torus, the other torus, two fixed torus. I have here P that is repelling, and Q that is an attracting point. And then, of course, if the derivative of phi is weak enough, this is a partially hyperbolic defiomorphism, and it is an action A. You have an attractor, a repeller that is hyperbolic, and you have here this torus, sorry, this torus, both of them, are saturated by unstable leaves and stable leaves. But this is not a robust situation. If you think a little bit, for the repeller, if you make up perturbation, almost all perturbation, you can break down this joint integrability between the stable and the unstable manifold. If you perturb, you will obtain that this, of course, is a repeller, and the stable is still in the repeller, it is contained in the repeller, but the unstable one is no longer contained in the repeller, and then you will obtain that this is outside and starts to go to the attractor. This with almost every small perturbation. Then that means that this is a very simple example. After the perturbation, you still have a torus here that is no longer differentiable, it's only a continuous torus. Why I like this example is because the spirit of the idea behind the torus is that when you have the basin of an attractor, it's very difficult, of course, the basin of attractors, whatever is the attractor, are saturated by a stable manifold. And then it's difficult to have unstable manifolds in the boundary of the basin of the attractor. This is the philosophy behind this. And then I will give another example, a more complicated example that is due to Cannes. It's very difficult, the original example, when you have the product, you have that in this torus, in the repeller, you have unstable manifolds, strong unstable manifolds. This torus is the boundary of the basin of attraction of this one. But this situation is not robust. If you perturb this, you don't have unstable manifolds in the boundary of the basin of attraction of these other torus. Yes, yes, yes, because you have, this is a product with this, everybody goes to the south pole, the torus that is in the south pole. This is the situation. Yes, yes, this one. This is not a robust situation. You can perturb and destroy the situation to have unstable manifolds in the boundary of the basin of these torus. You still have here a torus, but that is a repeller, but it doesn't contain any strong unstable manifolds. See, it varies continuously. You have the conjuacy, it's an axiom A, this is a hyperbolic set. And then you have a conjuacy with these dynamics. But it is no longer differentiable. This torus, this torus after perturbations will be only continuous or something. Continues to be in boundary, yes. But for instance, you cannot support here a give measure because you don't have the, the strong unstable manifolds. This is another example. Also, in the plot of the torus, time the circle, but this is a, this is more complicated. I will take something that is A and sign. The phi now depends on the base point. And I will make some, I will put some hypothesis in this phi. And I will suppose for simplicity that A has two fixed points. It's possible. It has two fixed points. Here it works. And I have here P and Q, these are P, zero, zero. And here I have the one-half, that is, and Q, one-half, okay? The first thing is that on this circle, we suppose that P is an attracting point and this is a repeller. And on this circle, we have the, in the other way, this is a repeller and this is an attractor, okay? These are north pole, south pole, the thermo things. Here, P is an attractor and here Q is a repeller. Q here is an attractor. P here is a repeller, okay? And of course, another condition is that the derivative of phi is small enough to be such a way that this is a partially hyperbolic thermomorphism. And the other thing is that if you integrate the derivative of phi at zero, at zero would reflect to the back measure here in the torus, sorry. This is smaller than zero. And the same for the other torus. So you can obtain a family of the thermomorphism like this with these properties. It's not very difficult. And then I have the following. I have two measures, sorry, M0 and one-half, two measures here that are SRE measures, physical measures, because with this condition you have that the center of the exponent is negative for both. And if you take here any stable manifold for this measure and you iterate it, it is dense in the whole torus, okay? Because you have that the stable manifold of this point is the stable manifold here in the torus times the circle minus this point. This is dense in the torus. And if you take here a passing stable manifold for this measure, it's a two-dimensional manifold. And then you have, since the unstable of this point is dense, it will intersect here and then you iterate this and we obtain that the orbit of this manifold is dense. And then the basin of attraction of this measure of this torus is dense, the whole torus. And the same for the other measure. It's the same argument with Q instead of F. Both measures have dense basins, okay? And this is what is called intermingulate basins, okay? And then we want to show that this type of phenomena are not robust, both, okay? These are, I just saw, I don't know. Yes, I think that here the union of the basins has full measure in this case. And then let me state our result. We have a partially hyperbolic defiomorphism of the torus, dynamically coherent. That means that we can integrate the center and with compact center lifts like this. And suppose that you have, you know, an SRV measure or physical measure, you know, a physical measure with a negative center point, center. There are two or one positive negative, always. And we have an invariant, of course compact, invariant to such a rated set K, contained where in the closure of the basin of the measure minus the support of the measure. Like this torus here, the repeller, and in the original example when you have the product, and like, what? These torus for this measure, and these are torus for this measure, okay? This is the K. Yes, on the torus, on the torus, yeah. Yes, positive volume, of course, because you have that here is Lebesgue, then it's a SRV, because the other exponent is negative. The center exponent is negative. The unstable manifolds are the strong and stable manifolds of the defiomorphism. And you have Lebesgue on the torus. The lines of the torus, then the measure is SRV. It's Lebesgue. Both are physical measures. In this case, yes, I don't care, but they are the only, for the result, I don't care if they are the only physical measures, but in this case they are. Mu is one of these, for instance this. Yes, mu, of course, is ergodic. Yes, that's the meaning of this. It's in the closure, because the boundary of the basin here is everything else. But because the basins are dense, but you take a set in the closure of the basin, but that doesn't intercept the support of the measure, that contains all its unstable dists. Strongest dists. Here is you, and here is mu. Let me change here. And the conclusion is that K contains torus tangent to ES plus U. This is not rost. We have a result with Keith, Riko, Tenhans, Riko, Hanna, Anna Thaleskaya, showing that accessibility is open and dense among the partially hyperbolic defects with one dimensional center. Then you cannot have this, something that integrates ES plus U for an open and dense set. That this is not rost is a corollary of this. And another comment is that, well, if the diffio is dynamically coherent, in the three torus is the only case where you can have this kind of situation, like this. Because if you have a partially hyperbolic diffiomorphism in the three torus, or it has, well, dynamically coherent, or all the center manifolds are circuits, compact, or it is semi-conjugated to an anosop diffiomorphism of the three torus, and in this case you have only one minimal set for the foliation, for the unstable foliation. And then you can have only, this is always usaturated, the support of a physical measure. And then you cannot have other set with the joint from this, because you have only one minimal set. Then in the three torus, this answers the problem of the robustness of this kind of phenomenon. Sorry? Yes, yes, yes, yes. It's only to, this is something, it's only to have this, because the measure is symbolic, and then if you have this, you obtain that the Lyapunov exponent is negative. This coincides with the, this integral is equal to the Lyapunov exponent for this measure. The measure is symbolic, and then you integrate the derivative in the center, which is absolutely continuous. Yes, it's absolutely continuous with respect to, if you take the conditional measures, here is the linear one, the conditional measures are the length in the line, and then this is physical, okay? Well, this was one, then in, well, in the three torus, we have also the non-dynamically coherent defiomorphism. I don't know how to deal with this case. And in other manifolds, in the nil manifolds, you have that also, there is only one minimal set for the stable, for any partial hyperbolic defiomorphism in three nil manifolds, you have only one minimal set for the unstable filiation, and then again the result is trivial. You cannot have, you don't have this type of examples. Well, and the proof, and the time may happen. The idea is, first we try to show that the intersection of K, we have K, and you have the intersection of K with a center leaf. K intersects every center leaf because you have the unstable leaves that are projected onto the unstable leaf of the anoso. If you take the quotient by the center, the dynamic is anoso, and the unstable leaves project onto the unstable leaves of this anoso, and then these are dense, and then since K is compact, the projection restricted to K is onto this tutorial. And then that means that K intersects all the center leaf, every center leaf intersects, and then you have your intersection of K with a center leaf. And then we show that this intersection is a final set. And the idea of the proof, this is in fact the delicate part because of course if you take, if the intersection is infinite, then you will have three points that are very close to each other, these three points, and then this is the intersection with K, then you have your unstable manifolds here that are the K unstable manifolds. And then if you take the stable manifolds through these unstable manifolds, you have three small disks, but this is in the closure of this, and then in the middle you have a point that is in the basin of the mesh. And the idea is that we have a problem here because I want to show that the stable manifold of this point intersects one of these sets. It's in the middle, you have three, but you cannot take the strong stable manifold because the strong stable manifold through this point is parallel to this disk. And then you have to show that the center manifold is long and goes to the support of the measure. And this is the hard part. If I have time I can tell how to obtain this because in general you may have that the stable manifold is very thin in the center direction. It's very small in the center direction. And this is the problem. We have to show that the center is large because if the center is large, you intersect here with it. It's large and it is contained in the stable manifold of this point. We want this, okay? This point is in the stable manifold of the, in the basin of the measure is in the stable manifold of some point of the support. We want to show that this center is contained in this stable manifold. And then if you have this intersection you have a contradiction because the points that are in this disk are not in the basin of the tractor because are stable manifolds of points of K that is invariant and then for the filter these points go to K, not to the support of the measure. And then this is more of proving that the center is long enough is the way to show that you have a finite number of points. And then if you have a finite number of points you manage to show that it's not completely trivial not difficult to show that the intersection, the number of points that intersect each center is constant, okay? And then if you have that this is constant you will have that this point but I continuously with the center, okay? And then you have something K is a finite number of points on each center that vary continuously with the center you are covering the two torus then this is a torus, okay? You locally will have here a disk at this point when you vary the center. And then, well, we have that this is a torus it is you saturated this torus and I want to show that it is S saturated, okay? And then what happens if it is not? If I have here the part of K of this torus here the unstable manifold and suppose that the stable manifold of this point is not contained in this set it leaves the set. Then by continuity all the in a neighborhood of this point you have that all stable manifolds do the same, okay? Very near here. And then you obtain an open set here a neighborhood of this point or something close to this point that is here where every point has an stable manifold that is this way, okay? But in this neighborhood all this stable manifold intersects K but in this neighborhood you have points of the basin because this is about in the closure of the basin and the stable manifold of this point cannot intercept K because I are in the basin of the measure and then this situation is impossible and then you have that this torus contains also the stable manifolds, okay? And then the more technical part for the end I want to show that the following that if I have a point X that is in the basin of nu and it changes the name of the measure then there is Y in the support of nu such that the arc, the center arc joining X or one center arc because you have two, the center is a circle one center arc joining X and Y is contained in the stable manifold of Y or... Here stable in the sense of basin the two dimensional manifold now the stable is impossible for the two dimensional stable manifold and then the idea is the following of course we have that X is in the stable manifold of some point X is in the stable manifold of C with C in the support of the measure okay? This is because it's in the basin and then of course I can suppose that C is in some passing block and you iterate it and after many iterations this is an iterate of C but I can suppose that X is very near C for the future I will take the iterate size this is fn of Z for some large n and this is fn of X this is here these are very close and we have the following the support of the measure is also usaturated and the intersection of a close usaturated set with the centers has some continuity properties it's semi-continuous in some sense because it's closed and then the limit is contained in the limit then you can suppose that very near here there is a point where this intersection in a continuous way it's a point of continuity of this intersection because the intersection is a semi-continuous function and then if you are very near W you have that all points here intersect in the centers if you take a point here near W you take that there is a point of the support in the center that is very near this point then the way to show this roughly speaking is to take the passing block of the point Z you intersect here with a small neighborhood and then you have here a set of positive measures where the stable manifolds are large because you are in a passing block and then you iterate this point again the point F and Z in such a way you go to this neighborhood and then you obtain that the center is very short and it's contained in the stable manifold because the stable manifold is large because you are in a passing block and you iterate for the past and obtain the property of having the center inside the stable manifold